such that \(M_i\ne M_{i+1}\) , is finite. In analogous way is defined Noetherian ring; a ring \(R\)  is Noetherian if it is Noetherian as a left module over itself. One can see equivalent definitions and properties e.g. in [1]}. In particular, every finitely generated module over a Noetherian ring is Noetherian. Since the ring of integers \(\mathbb {Z}\)  is an integral domain, as it is well known, it is Noetherian, and therefore, every finitely generated abelian group as a \(\mathbb {Z}-\) module is Noetherian.
